Nonlinear Sciences > Exactly Solvable and Integrable Systems

Title:
Commutation Relations and Discrete Garnier Systems

Abstract: We present four classes of nonlinear systems which may be considered discrete
analogues of the Garnier system. These systems arise as discrete isomonodromic
deformations of systems of linear difference equations in which the associated
Lax matrices are presented in a factored form. A system of discrete
isomonodromic deformations is completely determined by commutation relations
between the factors. We also reparameterize these systems in terms of the image
and kernel vectors at singular points to obtain a separate birational form. A
distinguishing feature of this study is the presence of a symmetry condition on
the associated linear problems that only appears as a necessary feature of the
Lax pairs for the least degenerate discrete Painlev\'e equations.